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>Now we modify the thought experiment a bit. We still have the hundred cubicles but this time they are not painted blue or red. Instead they are numbered from 1

>Now we modify the thought experiment a bit. We still have the hundred cubicles but this time they are not painted blue or red. Instead they are numbered from 1 to 100. The numbers are painted on the outside. Then a fair coin is tossed (by God perhaps). If the coin falls heads, one person is created in each cubicle. If the coin falls tails, then persons are only created in cubicles 1 through 10. >You find yourself in one of the cubicles and are asked to guess whether there are ten or one hundred people? Since the number was determined by the flip of a fair coin, and since you haven’t seen how the coin fell and you don’t have any other relevant information, it seems you should believe with 50% probability that it fell heads (and thus that there are a hundred people). The conclusion in the last sentence is incorrect is an incredibly subtle way. Since 10 times more people are in cubicles in the heads case, the probability that you find yourself in a cubicle at all is ten times higher in that case, which affects your prior. By Bayes' theorem: P(HEADS| WAKE UP) = (P(WAKE UP|HEADS) * P(HEADS)) / ( P(WAKE UP |HEADS) * P(HEADS) + P(WAKE UP|TAILS) * P(TAILS) ) = (10 * P(WAKE UP|TAILS) * P(HEADS)) / ( 10 * P(WAKE UP|TAILS)* P(HEADS) + P(WAKE UP|TAILS) * P(TAILS) ) = 10 / (10 + 1) = 10 / 11, or about 91% chance of heads given that you find yourself in a cubicle.

lisper 12y ago
> the probability that you find yourself in a cubicle at all is ten times higher in that case No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1.
- spacehome 12y ago
  The reason why this is so subtle is that it's not a given that you're in a cubicle. It's more likely for you to be put in a cubicle the more people who are put into cubicles.
  - baddox 12y ago
    > The reason why this is so subtle is that it's not a given that you're in a cubicle. The way the problem is presented, it is a given that you're in a cubicle. If you wanted to be more explicit about it, you could say that God, before creating humanity, flipped a coin, and then created either 10 or 100 humans and placed them into cubicles. You wake up and have to estimate the probability of that coin flip.
    - bluecalm 12y ago
      It is given that you are but it's not given that that was a sure thing to happen. Imagine this: God flips a coin, creates 10 cubicles. If the coin is heads he puts you in one of them if it tails the God flips another coin and if it tails again he puts you in one of the cubicles but if it heads he doesn't. You wake up in a cubicle. It's given that you are in one! what are the chances the first coin landed heads ? The probability isn't 50% because there is underlying probability that you would be in cubicle at all (1 vs 0.5 after first flip). There is the same problem with doomsday argument just better masked.
      - lisper 12y ago
        Ah, I see. It makes more sense if you forget about cubicles and just call the final outcomes A or B. Call the outcomes of the two coin flips H1/T1 and H2/T2. We have: H1->A, T1,H1->A, T1,T2->B. P(H1|A) is therefore 67% if the coin is fair, but in any event P(H1|A)>P(H1). So the doomsday argument is actually correct. :-) But it's really nothing more than the (IMO vacuous) observation that (say) half of the people who ever live will have a birth order number greater than the median, and so the odds are 50-50 that fewer people will be born after you than before you. More generally, the odds that N times as many people will be born after you as before you is equal to 1/2N.
- bluecalm 12y ago
  >>No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1. But it's not 1 that you would be in a cubicle at all. The paradox disappears if the problem was formulated like this: "God shuffles DNA and there is 0.0000000000001 chances tha it shuffles up you and he does that for every cuebicle". Or: "God create you first, then flips a coin and put you at random in one of the created cubicles". Formulation of the problem should mention which one is it. If it doesn't we are back to guessing what God does (similarly to 2 envelope problem which comes down to guessing what the sponsor's preferences for amounts are). As there are infinitely many ways God could decide to create humans in cubicles you can't answer that without giving some guess (priori) for probability distribution over those choices.
  - spacehome 12y ago
    There are only finitely many ways to arrange matter in a finite volume under a finite temperature. If it's safe to assume that anything we could classify as "human" will fit inside a sphere of radius 1 light-year and has a temperature under 10^20 K, there are finitely many humans.
    - bluecalm 12y ago
      Yeah but it's not given that God chooses at random who is going to cubicle. He might have for example created you first and then put you in a cubicle regardless if it's 100 of them or 10. I agree with your resolution of that paradox though. It's reasonable to assume - having no information about the process - that God creates humans randomly and it's 10x more likely that you are created in a world with 100 cubicles than in a world in 10 cubicles. In my mind there is no paradox at all there.
  - otakucode 12y ago
    And I don't understand why we ignore that human beings created the cubicles and there is no random god involved in any of it. Whether there is a human in the cubicle or not is entirely a result of physical processes. Any statement about probability is nothing but an explanation of our ignorance. Yeah, we might not be capable of knowing the totality of physical history of all the people who build the cubicles and where they are now and such... so what? That says nothing about the history of their species.
AlexMennen 12y ago
This reasoning uses the self-indication assumption, which I tentatively also agree with, but there are some pretty good criticisms of it. http://www.anthropic-principle.com/preprints/olum/sia.pdf http://www.anthropic-principle.com/preprints/olum/sia.pdf

lisper 12y ago

> the probability that you find yourself in a cubicle at all is ten times higher in that case No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1.

spacehome 12y ago
The reason why this is so subtle is that it's not a given that you're in a cubicle. It's more likely for you to be put in a cubicle the more people who are put into cubicles.
- baddox 12y ago
  > The reason why this is so subtle is that it's not a given that you're in a cubicle. The way the problem is presented, it is a given that you're in a cubicle. If you wanted to be more explicit about it, you could say that God, before creating humanity, flipped a coin, and then created either 10 or 100 humans and placed them into cubicles. You wake up and have to estimate the probability of that coin flip.
  - bluecalm 12y ago
    It is given that you are but it's not given that that was a sure thing to happen. Imagine this: God flips a coin, creates 10 cubicles. If the coin is heads he puts you in one of them if it tails the God flips another coin and if it tails again he puts you in one of the cubicles but if it heads he doesn't. You wake up in a cubicle. It's given that you are in one! what are the chances the first coin landed heads ? The probability isn't 50% because there is underlying probability that you would be in cubicle at all (1 vs 0.5 after first flip). There is the same problem with doomsday argument just better masked.
    - lisper 12y ago
      Ah, I see. It makes more sense if you forget about cubicles and just call the final outcomes A or B. Call the outcomes of the two coin flips H1/T1 and H2/T2. We have: H1->A, T1,H1->A, T1,T2->B. P(H1|A) is therefore 67% if the coin is fair, but in any event P(H1|A)>P(H1). So the doomsday argument is actually correct. :-) But it's really nothing more than the (IMO vacuous) observation that (say) half of the people who ever live will have a birth order number greater than the median, and so the odds are 50-50 that fewer people will be born after you than before you. More generally, the odds that N times as many people will be born after you as before you is equal to 1/2N.
bluecalm 12y ago
>>No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1. But it's not 1 that you would be in a cubicle at all. The paradox disappears if the problem was formulated like this: "God shuffles DNA and there is 0.0000000000001 chances tha it shuffles up you and he does that for every cuebicle". Or: "God create you first, then flips a coin and put you at random in one of the created cubicles". Formulation of the problem should mention which one is it. If it doesn't we are back to guessing what God does (similarly to 2 envelope problem which comes down to guessing what the sponsor's preferences for amounts are). As there are infinitely many ways God could decide to create humans in cubicles you can't answer that without giving some guess (priori) for probability distribution over those choices.
- spacehome 12y ago
  There are only finitely many ways to arrange matter in a finite volume under a finite temperature. If it's safe to assume that anything we could classify as "human" will fit inside a sphere of radius 1 light-year and has a temperature under 10^20 K, there are finitely many humans.
  - bluecalm 12y ago
    Yeah but it's not given that God chooses at random who is going to cubicle. He might have for example created you first and then put you in a cubicle regardless if it's 100 of them or 10. I agree with your resolution of that paradox though. It's reasonable to assume - having no information about the process - that God creates humans randomly and it's 10x more likely that you are created in a world with 100 cubicles than in a world in 10 cubicles. In my mind there is no paradox at all there.
- otakucode 12y ago
  And I don't understand why we ignore that human beings created the cubicles and there is no random god involved in any of it. Whether there is a human in the cubicle or not is entirely a result of physical processes. Any statement about probability is nothing but an explanation of our ignorance. Yeah, we might not be capable of knowing the totality of physical history of all the people who build the cubicles and where they are now and such... so what? That says nothing about the history of their species.

AlexMennen 12y ago

This reasoning uses the self-indication assumption, which I tentatively also agree with, but there are some pretty good criticisms of it. http://www.anthropic-principle.com/preprints/olum/sia.pdf http://www.anthropic-principle.com/preprints/olum/sia.pdf